one-way ANOVA?

I was wondering if anyone could help me out. I want to compare the means of three groups. I hypothesize that there will be no significant difference in the means between two of the groups but a significant difference between these two groups and a third group. I thought this would be a one-way ANOVA but it does not show a significant difference between groups. However, if I do three separate t-tests I see the trend I was expecting. Considering I would have to adjust my alpha to account for the multiple tests (which would be fine, all tests were p<0.0001 but it is awfully sloppy) and thought this is a very simple comparison that I should be able to figure out, I was wondering if anyone could tell me what I'm doing wrong.
Thank you!! Happy holidays
You have to do two different hypothesis tests. There is no easy way in mathematical statistics I think to write the test you speak of. You need to do 3 hypothesis tests: u1=u2, u1=u3, and u2=u3. It sounds like you've done them already.

Your other option is to do Ho: u1=u2=u3, Ha: At least two not equal If you do a combined test and they are considered equal, then that's that. However you should consider the pooled alpha of the test.